Part 654 Stream Restoration Design National Engineering Handbook

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1 United States Department of Agriculture Natural Resources Conservation Service Stream Restoration Design Chapter 6

2 Issued August 007 Cover photo: Stream hydraulics focus on bankfull frequencies, velocities, and duration of flow, both for the current condition, as well as the condition anticipated with the project in place. Effects of vegetation are considered both in terms of protection of the bank materials, as well as on changes in hydraulic roughness. Advisory Note Techniques and approaches contained in this handbook are not all-inclusive, nor universally applicable. Designing stream restorations requires appropriate training and experience, especially to identify conditions where various approaches, tools, and techniques are most applicable, as well as their limitations for design. Note also that product names are included only to show type and availability and do not constitute endorsement for their specific use. The U.S. Department of Agriculture (USDA) prohibits discrimination in all its programs and activities on the basis of race, color, national origin, age, disability, and where applicable, sex, marital status, familial status, parental status, religion, sexual orientation, genetic information, political beliefs, reprisal, or because all or a part of an individual s income is derived from any public assistance program. (Not all prohibited bases apply to all programs.) Persons with disabilities who require alternative means for communication of program information (Braille, large print, audiotape, etc.) should contact USDA s TARGET Center at (0) (voice and TDD). To file a complaint of discrimination, write to USDA, Director, Office of Civil Rights, 400 Independence Avenue, SW., Washington, DC , or call (800) (voice) or (0) (TDD). USDA is an equal opportunity provider and employer. (0 VI NEH, August 007)

3 Contents Purpose Introduction 6 (a) Hydraulics as physics...6 (b) Hydraulics as empiricism Channel cross-sectional parameters Dimensionless ratios 6 4 (a) Froude number (b) Reynolds number Continuity Energy Momentum Specific force Stream power Hydraulic computations 6 9 (a) Uniform flow (b) Determining normal depth (c) Determining roughness coefficient (n value)...6 (d) Friction factor (e) Accounting for velocity distributions in water surface profiles (f) Determining the water surface in curved channels (g) Transverse flow hydraulics and its geomorphologic effects (h) Change in channel capacity Water surface profile calculations 6 5 (a) Steady versus unsteady flow (b) Backwater computational models Weir flow Hydraulic jumps Channel routing 6 (a) Movement of a floodwave...6 (b) Hydraulic and hydrologic routing...6 (c) Saint Venant equations...6 (d) Simplifications to the momentum equation Hydraulics input into the stream design process 6 5 (a) Determining project scope and level of analysis (b) Accounting for uncertainty and risk (0 VI NEH, August 007) 6 i

4 Tables Table 6 Froude numbers for types of hydraulic jumps 6 0 Table 6 Project dimensions by type and stage of project 6 5 Table 6 Scope of hydraulic analyses by project type 6 5 Figures Figure 6 Channel cross-sectional parameters 6 Figure 6 Specific energy vs. depth of flow 6 6 Figure 6 Problem cross section 6 0 Figure 6 4 HEC RAS screen shot for uniform flow computation 6 Figure 6 5 Cross-sectional dimensions 6 Figure 6 6 Looking upstream from left bank 6 Figure 6 7 Looking downstream on right overbank 6 Figure 6 8 Sand channel cross section 6 4 Figure 6 9 Plot of flow regimes resulting from stream power 6 5 vs. median fall diameter of sediment Figure 6 0 General bedforms for increasing stream power 6 6 Figure 6 Flow velocities for a typical cross section 6 8 Figure 6 Spiral flow characteristics for a typical reach 6 0 Figure 6 Flow characteristics for a typical reach 6 0 Figure 6 4 Channel centerline at centroid of flow 6 Figure 6 5 Point bar development 6 Figure 6 6 Seasonal hydrograph 6 4 Figure 6 7 Standard step method 6 5 Figure 6 8 Problem determinations for determination of log 6 6 weir 6 ii (0 VI NEH, August 007)

5 Figure 6 9 Profile for crest of log weir problem 6 6 Figure 6 0 Determination of jump length based on upstream 6 Froude number Figure 6 Parameters involved with modeling a hydraulic 6 jump (0 VI NEH, August 007) 6 iii

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7 Purpose Introduction Human intervention in the stream environment, especially with projects intended to restore a stream ecosystem to some healthier state, must fully consider the stream system, stream geomorphology, stream ecology, stream hydraulics, and the science and mechanics of streamflow. This chapter provides working professionals with practical information about hydraulic parameters and associated computations. It provides example calculations, as well as information about the role of hydraulic engineers in the design process. The hydraulic parameters used to evaluate and quantify streamflow are described in this chapter. The applicability of the various hydraulic parameters in planning and design in the stream environment is presented. The complexity of streamflow is addressed, as well as simplifying assumptions, their validity, and consequences. Guidance is provided for determining the level of analysis commensurate with a given project s goals and the associated hydraulic parameters. Finally, a range of analytical tools is described, the application of which depends on the complexity of the project. Stream hydraulics is a complex subject, however, and this chapter does not provide exhaustive coverage of the topic. Readers are encouraged to supplement this information with the many good references that are available. Stream hydraulics is the combination of science and engineering for determining streamflow behavior at specific locations for purposes including solving problems that generally originate with human impacts. A location of interest may be spatially limited, such as at a bridge, or on a larger scale such as a series of channel bends where the streambanks are eroding. Flood depth, as well as other hydraulic effects, may need to be determined over long stretches of the channel. An understanding of flowing water forms the basis for much of the work done to restore streams. The discipline of hydrology involves the determination of flow rates or amounts, their origin, and their frequency. Hydraulics involves the mechanics of the flow and, given the great power of flowing water, its affect on bed, banks, and structures. A stream is a natural system that constantly adjusts itself to its environment and participates in a cycle of action and reaction. These adjustments may be gradual, less noticeable, and long term, or they may be sudden and attention grabbing. The impacts causing a stream to react may be natural, such as a rare, intense rainfall, or human-induced, such as the straightening of a channel or filling of a wetland. However, the reaction of a stream to either kind of change may be more than localized. A stream adjusts its profile, slope, sinuosity, channel shape, flow velocity, and boundary roughness over long sections of its profile in response to such impacts. After an impact, a stream may restore a state of equilibrium in as little as a week, or it may take decades. (a) Hydraulics as physics Stream characteristics are derived from the basic physics of flowing water. Fluid mechanics is an old science with well-established physical relationships. Typically, simple empirical equations are used that do not account for all the variability that occurs in the flow. An example is Bernoulli s equation for balancing flow depth, velocity, and pressure. In this case, the flow must be considered steady. If it is important to assess how flow depth, velocity, and/or pressure (0 VI NEH, August 007) 6

8 change over time, Bernoulli s equation by itself will not be sufficient. The assumption that flow velocity is generally downstream in direction is also a common simplification in the analysis of streamflow. Real streams have many eddies where the flow circulates horizontally. Streams also have areas of upwelling, roiling, and vertical circulation. While designers commonly make use of an average velocity at a given cross section, the actual velocities in the plane of a cross section vary markedly from top to bottom, side to side, and in direction, varying with time and three-dimensional space. redirecting a stream and sometimes transported downstream. Sediment transport is influenced by velocity vectors near the water/sediment boundary, and these bed velocities may not be well predicted by an average cross-sectional velocity. Many of the analytical sediment predictive techniques include many empirical estimates of specific parameters. More information on the analytical, as well as empirical approaches to sediment transport, is provided in other chapters of this handbook. More information on sedimentation analysis is provided in NEH and NEH654.. Water surface profile analyses generally assume a constant flow elevation across a given cross section. Real streams, however, super-elevate their water surfaces in curved channel sections and may set up significant surface wave patterns that defy prediction. Finally, hydraulic analyses often assume that water flows against a fixed boundary. Real streams actually readjust their bed and banks constantly, move significant amounts of sediment, and transport unpredictable amounts of natural or humanmade debris. It is, therefore, important to understand the limitations and restrictions of any equations before using them to obtain necessary information. (b) Hydraulics as empiricism Although thoroughly founded in physics, many hydraulic relationships require empirical coefficients to account for unmeasured or estimated processes. One of the parameters that has a significant influence on hydraulic calculations is surface roughness, in the form of Manning s n value, the Chézy C, or the Darcy-Weisbach friction factor. While the Darcy-Weisbach friction factor is generally considered to be more theoretically based, Manning s n is more commonly used for most stream design and restoration analysis. Roughness is a function of many stream physical properties including bed sediment size, vegetation, channel sinuosity, channel irregularity, and suspended sediment load. As a result, many of the estimates have inherent degrees of empiricism in their estimate. Sediment transport also requires empirical input. Sediment particles vary in size and properties, from tiny silt particles that adhere to large boulders, sometimes 6 (0 VI NEH, August 007)

9 Channel crosssectional parameters A variety of channel cross-sectional parameters are used in the hydraulic analysis of streams and rivers. It is important to measure and use these parameters consistently and accurately. A generalized cross section is shown in figure 6. The flow depth is the distance between the channel bottom and the water surface. For rectangular channels, the depth is the same across an entire cross section, but it obviously varies in natural channels. Depth is often measured relative to the channel thalweg (or lowest point). Normal depth is the depth of flow in a uniform channel for which the water surface is normal or parallel to the channel profile and energy slope. For a cross section aligned so that streamlines of flow are perpendicular, the flow area is the area of the cross section between bed and banks and water surface. For a rectangular channel, flow area is depth multiplied by top width. For a natural channel cross section, the area may be approximated with the sum of trapezoidal areas between cross-sectional points. The top width of a channel cross section at the water surface, typically designated as T, is a factor in the hydraulic depth. The hydraulic depth is the ratio of the cross-sectional area of flow to the free water surface or top width. The hydraulic depth, d, is generally used either in computing the Froude number or in computing the section factor for critical depth. Since only one critical depth is possible for a given discharge in a channel, the section factor, Z, can be used to easily determine it (Chow 959). Z = A d (eq. 6 ) Qcritical = Z g (eq. 6 ) For a cross section normal to the direction of flow, the wetted perimeter (typically designated P) is the length of cross-sectional boundary between water and bed and banks. The hydraulic radius is the ratio of the cross-sectional area of flow to the wetted perimeter or flow boundary. The hydraulic radius, R= A/P, is used in Manning s equation for calculation of normal depth discharge, as well as for calculation of shear velocity. Velocity is a physics term for a change in distance during a time interval. Flow velocity refers to the areal extent of the flow (in a cross section) for which a velocity is specified. For example, an average velocity that applies to an entire cross-sectional area may be determined from V = Q/A or if the discharge is unknown, a uniform flow velocity may be determined from Manning s equation. Figure 6 Channel cross-sectional parameters (per ft of channel length) Top width (T) Cross-sectional area (A) Depth (d max ) Wetted perimeter (P) Thalweg (lowest point of the channel) (0 VI NEH, August 007) 6

10 Another useful formulation is critical velocity, which is average flow velocity at critical depth, and is calculated from equation 6 : V V cr = critical velocity g = gravitational acceleration d cr = critical depth cr = gd (eq. 6 ) cr Determining the state of flow is a matter of determining whether the velocity is greater than critical velocity V cr (supercritical flow) or less than critical velocity V cr (subcritical flow). Conveyance is a measure of the flow-carrying capacity of a cross section which is directly proportional to discharge. Conveyance, typically designated K, may be expressed from Manning s equation (without the slope term) as: or K = 486. AR n A = flow area (ft ) R = hydraulic radius (ft) Q = flow rate (ft /s) S = slope, dimensionless (eq. 6 4a) Q K = (eq. 6 4b) S In backwater calculations, change in conveyance from cross section to cross section is a useful way to determine the adequacy of section spacing in a stream reach. Within a cross section, conveyance may be used to compare channel and overbank flow carrying capacity Dimensionless ratios Dimensionless ratios (also referred to as dimensionless numbers) are used to provide information on flow condition. The units of the variables used in the equation for a dimensionless ratio are such that they cancel. The two most commonly used ratios are Froude and Reynolds numbers. Being dimensionless allows their application to be made across a variety of scales. (a) Froude number The Froude number is a dimensionless ratio, relating inertial forces to gravitational forces. The Froude number represents the effect of gravity on the state of flow in a stream (Chow 959). This useful number was derived by a nineteenth century English scientist, William Froude, who studied the resistance of ships being towed in water. He observed wave patterns along the hull of a moving ship and found that the same number of waves would occur as long as the ratio of the ship s speed to the square root of its length were the same. Applied in hydraulics, the length is replaced by hydraulic depth, as shown in equation 6 5. V F = (eq. 6 5) gd V = velocity (ft/s) g = acceleration due to gravity (. ft/s ) d = flow depth (ft) If the Froude number is less than one, gravitational forces dominate and the flow is subcritical, and if greater than one, inertial forces dominate and the flow is supercritical. The Froude number is used to determine the state of flow, since, for subcritical flow the boundary condition is downstream, and for supercritical flow it is upstream. When the Froude number equals one, the flow is at the critical state. (b) Reynolds number The Reynolds number is also a dimensionless ratio, relating the effect of viscosity to inertia, used to determine whether fluid flow is laminar or turbulent (Chow 959). The Reynolds number relates inertial forces to 6 4 (0 VI NEH, August 007)

11 viscous forces and was derived by a nineteenth century English scientist, Osborne Reynolds, for use in wind tunnel experiments. Inertia is represented in equation 6 6 by the product of velocity and hydraulic radius, divided by the kinematic viscosity of water, with units of length squared per time. For turbulent flow Re>000, for laminar, Re<500, and values between these limits are identified as transitional. Re = VR ν V = velocity (ft/s) R = hydraulic radius (ft) ν = kinematic viscosity (ft /s) (eq. 6 6) For use in sediment transport analysis, the Reynolds number has been formulated to apply at the watersediment boundary. In this case, the velocity is local to the boundary and termed shear velocity (V * ). Also, the length term is not the hydraulic radius, but roughness height, or the diameter of particles (D) forming the boundary. This boundary Reynolds number has also been called the bed Reynolds number or shear Reynolds number Continuity Open channel flow has a liquid surface that is open to the atmosphere. This boundary is not fixed by the physical boundaries of a closed conduit. Water is essentially an incompressible fluid, so it must increase or decrease its velocity and depth to adjust to the channel shape. If no water enters or leaves a stream (a simplification that can be made over short distances) the quantity of the flow will be the same from section to section. Since the flow is incompressible, the product of the velocity and cross-sectional area is a constant. This conservation of mass can be written as the continuity equation as follows: Q = VA (eq. 6 8) While the continuity equation can be used with any consistent set of units, it is normally expressed as: Q = quantity of flow (ft /s) A = cross-sectional area (ft ) ) V = average velocity (ft/s) V D Re V * = boundary shear velocity (ft/s) D = particle diameter ν = kinematic viscosity (ft /s) * bed = (eq. 6 7) ν Because streamflow is almost exclusively turbulent, the Reynolds number is not needed as a flag of turbulence. The Reynolds number has value for sedimentation analyses in that drag coefficients have been empirically related to Reynolds number. Another important use in sedimentation involves incipient motion of sediment particles. Studies have related the bed Reynolds number to critical shear stress (the initiation point of sediment movement). Through the Shields diagram, for example, one can determine critical shear, given a bed Reynolds number. Additional information on this topic is provided in NEH654.. (0 VI NEH, August 007) 6 5

12 Energy Energy, an abstract quantity basic to many areas of physics, is a property of a body or physical system that enables it to move against a force. It is an expression of work, which is force applied over a distance. Energy is the amount of work required to move a mass through a distance. Or, it is the amount of work a physical system is capable of doing, in changing from its actual state to some specified reference state. Many useful concepts of energy exist, the primary one being that, in a closed system, the total energy is constant, the concept of conservation of energy. Water energy is comprised of a number of components, often called head and expressed as a vertical distance. The potential energy of water, or pressure head, is a result of its mass and the Earth s gravitational pull. The kinetic energy of water is related to its movement and is called the velocity head. The Bernoulli equation (eq. 6 9) is an expression of the conservation of energy. z V V + y + α = z + y + α + h g g L (eq. 6 9) channels, the channel slope is sufficiently gradual for this angle to be small enough to be ignored. However, in slopes that are greater than 0 percent, this may become an issue that should be addressed. Another assumption is that flow is always perpendicular to the cross sections. Finally, alpha (α) in the equation is the energy coefficient, and it varies with the uniformity of velocity vectors in the cross section. For a fairly uniform velocity, alpha may be taken to be one. If velocity varies markedly over the cross section, alpha may go as high as. in sections of sudden expansion or contraction (Chow 959). Specific energy is a particular concept in hydraulics defined as the energy per unit weight of water at a given cross section with respect to the channel bottom. As shown in figure 6, specific energy can be helpful in visualizing flow states of a stream. The points d and d are alternate depths for the same energy level. Only one depth exists at the critical state, which is the lowest possible energy level for a given discharge. In natural streams, this is an unstable state since a very This expression shows the interrelationship of these energy terms, between two cross sections ( and ). Each term represents a form of energy, with depth y representing potential energy, the velocity term V representing kinetic energy, and z, a potential energy term relating all to a common datum in a plane perpendicular to the direction of gravity. The head loss or h L term is called a loss because any energy consumed between the two cross sections must be made up for by a change in height (or head). The head loss is the energy consumed by boundary friction, turbulence, eddies, or sediment transport. The velocity term represents velocity head and the depth term the pressure head. Figure 6 Specific energy vs. depth of flow Range of subcritical flow Although energy is a scalar quantity, without direction, the concept of energy as head has an orientation in the direction of gravity. Pressure, however, represents the magnitude of a force in the direction of whatever surface it impinges. So, as a channel slope steepens, the orientation of the pressure head is technically moving further from vertical. It is represented by the depth times the cosine of the slope angle. For most natural Depth d d Specific energy Critical state Range of supercritical flow Q higher than Q A given flow rate, Q Q lower than Q 6 6 (0 VI NEH, August 007)

13 small change in energy results in a relatively significant undulating change in depth. An understanding of flow energy is fundamental in hydraulic modeling. The specific energy at any cross section for a channel of small slope (most natural channels) and α = is: V E = y + g (eq. 6 0) Momentum In basic physics, momentum is the mass of a body times its velocity and is a vector quantity, whereas energy is scalar, lacking a direction. In hydraulics, the use of this concept is due mainly to the implication of Newton s second law, that the resultant of all forces acting on a body causes a change in momentum. The momentum equation in hydraulics is similar in form to the energy equation and, when applied to many flow problems, can provide nearly identical results. However, knowledge of fundamental differences in the two concepts is critical to modeling certain hydraulic problems. Conceptually, the momentum approach should be thought of as involving forces on a mass of flowing water, instead of the energy state at a particular location. Friction losses in momentum relate to the force resistance met by that mass with its boundary, whereas in the energy concept, losses are due to internal energy dissipation (Chow 959). The momentum equation can have advantages in modeling flow over weirs, drops, hydraulic jumps, and junctions, where the predominate friction losses are due to external forces, rather than internal energy dissipation. Interpreted for open channel, Newton s second law states that the rate of momentum change in this short section of channel equals the sum of the momentum of flow entering and leaving the section and the sum of the forces acting on the water in the section. Since momentum is mass times velocity, the rate of change of momentum is the mass rate of change times the velocity. The momentum equation may be written considering a small mass or slug of flowing water between two sections and and the principle of conservation of momentum. ( ) = + ρq β V β V P P W sin θ F fr (eq. 6 ) The left side of the equation is the momentum entering and leaving, and the right side is the pressure force at each end of the mass, with Wsinθ being the weight of the mass, θ being the angle of the bottom slope of the channel, and F fr being the resistance force of friction on the bed and banks. (0 VI NEH, August 007) 6 7

14 Specific force Specific force is the horizontal force of flowing water per unit weight of water. It is derived from the momentum equation. A specific force curve looks similar to the specific energy curve. The critical depth occurs both at the minimum energy for a given discharge and also at the minimum specific force for a given discharge. This similarity shows how energy concepts and force or momentum concepts can be employed similarly in many hydraulic analyses, often with nearly identical results. The designer should know what circumstances would cause the two approaches to diverge, however. Specific force concepts are applied over short horizontal reaches of channel, where the difference in external friction forces and force due to the weight of water are negligible. Examples are the flow over a broad-crested weir through a hydraulic jump or at junctions. One way to conceptualize why a momentum-based method, rather than an energy-based method, might be more applicable would be to energy changes in a hydraulic jump. Much energy is lost through turbulence caused by moving mass colliding with other mass that is not accounted for by energy principles alone. that becomes: Q Q + Ay = + A y (eq. 6 4) ga ga For a channel section of any other shape, the resultant pressure may be taken at the centroid of the flow area, at a depth, z, from the surface. Then the momentum formulation is: Q ga Q + Az = + A z (eq. 6 5) ga Either side of this equation is the definition of specific force, and the specific force is constant over a short stretch of channel such as a hydraulic jump. The first term represents change in momentum over time, and the second term the force of the water mass. As Chow (959) explains, specific force is sometimes called force plus momentum or momentum flux. An equation for specific force may be derived from the momentum equation. If the practitioner wishes to apply this equation to short sections of channel such as a weir or hydraulic jump, the frictional resistance forces, F fr can be neglected. With a flat channel of low slope, θ approaches 0, then the last two terms in equation 6 can be dropped. As a result, equation 6 becomes: ( ) = (eq. 6 ) ρq β V β V P P Assume also that the Boussinesq coefficient (β) is. From the fact that the pressure increases with depth to the maximum of ρgy at the channel bottom (y being depth, b being channel width, and ρ being fluid density), the overall pressure on the vertical flow area may be expressed as /ρgby. The velocities may be expressed as Q/A. For a rectangular channel: ρq Q A Q ρg A y A y A = ( ) (eq. 6 ) 6 8 (0 VI NEH, August 007)

15 Stream power Hydraulic computations Stream power is a geomorphology concept that is a measure of the available energy a stream has for moving sediment, rock, or woody material. For a cross section, the total stream power per unit length of channel may be formulated as: Ω = γqs = γvwds γ = unit weight of water (lb/ft ) Q = discharge (ft /s) S f = energy slope (ft/ft) v = velocity (ft/s) w = channel width (ft) d = hydraulic depth (ft) f f (eq. 6 6) English units are pounds per second per foot of channel length. A second formulation, unit stream power, is the stream power per unit of bed area: τ = bed shear stress v = average velocity Ω = τ 0 v (eq. 6 7) A third formulation relates stream power per unit weight of water: Ω = S v f (eq. 6 8) where the terms are as previously defined. (a) Uniform flow Water flowing in an open channel typically gains kinetic energy as it flows from a higher elevation to a lower elevation. It loses energy with friction and obstructions. Uniform flow occurs when the gravitational forces that are pushing the flow along the channel are in balance with the frictional forces exerted by the wetted perimeter that are retarding the flow. For uniform flow to exist: Mean velocity is constant from section to section. Depth of flow is constant from section to section. Area of flow is constant from section to section. Therefore, uniform flow can only truly occur in very long, straight, prismatic channels where the terminal velocity of the flow is achieved. In many cases, the flow only approaches uniform flow. Since uniform flow occurs when the gravitational forces are exactly offset by the resistance forces, a resistance equation can be used to calculate a velocity. The most commonly used resistance equation is Manning s equation (eq. 6 9). Q = 486. AR S n (eq. 6 9) given Q = VA then V = 486. R S n A = flow area (ft ) R = hydraulic radius (ft) S = channel profile slope (ft/ft) n = roughness coefficient (eq. 6 0) The.486 exponent is replaced by.0 if SI units are used. The flow area (A) and the hydraulic radius (R) relate how the flow interacts with the boundary. (0 VI NEH, August 007) 6 9

16 A rough estimate of the flow capacity or average velocity at a natural cross section may be determined with Manning s equation. A designer may assume a roughly trapezoidal cross section, estimating bottom width, side slopes, and profile slope from topographic maps. The roughness coefficient is a significant factor, and its determination is described in NEH (c). (b) Determining normal depth Normal depth calculation is one of the most commonly used analyses in stream restoration assessment and design. Several spreadsheets, computer programs, and nomographs are available for use in calculating normal depth. In a natural channel, with a nonuniform cross section, reliability of the normal depth calculation is directly related to the reliability of the input data. Sound engineering judgment is required in the selection of a representative cross section. The cross section should be located in a uniform reach where flow is essentially parallel to the bank line (no reverse flow or eddies). This typically occurs at a crossing or riffle. Determination of the average energy slope can be difficult. If the channel cross section and roughness are relatively uniform, surface slope can be used. Thalweg slopes and low-flow water surface slopes may not be representative of the energy slope at design flows. Slope estimates should be made over a significant length of the stream (a meander wavelength or 0 channel widths). Hydraulic roughness is estimated based on field observations and measurements. In addition to normal depth for a given discharge, these same procedures may be used to estimate average velocities in the cross section. These calculations do not account for backwater in a channel reach. The following example calculation refers to the cross section shown in figure 6. Example problem: Normal depth rating curve calculation Problem : Calculate a normal depth rating curve for each foot of depth up to 5 feet. Assume channel slope = and an n value = 0.0 Solution: For the value Q = AR S 49 n n S =. A and P need to be determined. R = A P A = 0 = 5 ft. ( ) = P 0 5 = ft R = ft ( ) + ( ) = ft A = ( ) = ft P 0 5 = + + R =. 9 ft ( ) + ( ) = ft A = ( ) = ft P 0 5 = + + R =. 07 ft, Figure 6 Problem cross section 0 ft ( ) + ( ) = ft A 4 = ( ) = ft P = + + R 4 =. 69 ft ( ) + ( ) = ft A 5 = Five -ft increments 5 ft 6 0 (0 VI NEH, August 007)

17 ( ) = ft P = + + R 5 =. 06 ft Solving for Q, then: Q Q Q Q Q = ( ) = = ft /s (at d ft) = ( ) = = ft /s (at d ft) = ( ) = = ft /s (at d ft) = ( ) = = ft /s (at d 4 ft) = ( ) = = ft /s (at d 5 ft) Problem : Determine the normal depth for a discharge of 50 cubic feet per second and the associated average velocity. Solution: From the rating curve calculated above, the 50 cubic feet per second discharge in this problem will be between Q and Q 4. A straight-line interpolation gives a depth of.4 feet. For velocity, since Q = VA V = 50 (. 4. 4) + 8(. 4) 4 ( ) =. 6 ft/s Discussion: The more complicated a section becomes, the more tedious is this hand calculation. Numerous computer programs, such as HEC RAS (USACE 00b), can perform normal depth calculations for a cross section of many coordinate points. A typical image from HEC RAS is shown as figure 6 4. Figure 6 4 HEC RAS screen shot for uniform flow computation (0 VI NEH, August 007) 6

18 (c) Determining roughness coefficient (n value) The roughness coefficient, an empirical factor in Manning s equation, accounts for frictional resistance of the flow boundary. Estimating this flow resistance is not a simple matter. This parameter is used in computation of water surface profiles and estimation of normal depths and velocities. Boundary friction factors must be chosen carefully, as hydraulic calculations are significantly influenced by the n choice. Factors affecting roughness include ground surface composition, vegetation, channel irregularity, channel alignment, aggradation or scouring, obstructions, size and shape of channel, stage and discharge, seasonal change, and sediment transport. Significant guidance exists in the literature regarding roughness estimation. Chow (959) discusses four general approaches for roughness determination. The U.S. Geological Survey (USGS) (Arcement and Schneider 990) published an extensive step-by-step guide for determination of n values. NRCS guidance for channel n value determination is available from Faskin (96). Finally, when observed flow data and stages are known, manual calculations or a computer program such as HEC RAS may be used to determine n values. With the many factors that impact roughness, and each stream combining different factors to different extents, no standard formula is available for use with measured information. As stated in Chow (959):...there is no exact method of selecting the n value. At the present stage of knowledge [959], to select a value of n actually means to estimate the resistance to flow in a given channel, which is really a matter of intangibles. To veteran engineers, this means the exercise of sound engineering judgment and experience; for beginners, it can be no more than a guess, and different individuals will obtain different results. Estimates of channel roughness may be made using photographs or tables provided by Chow (959), Brater and King (976), Faskin (96), and Barnes (967). NEH 5 supplement B, Hydraulics, can also be used to estimate roughness values. As roughness can change dramatically between surfaces within the same cross section, such as between channel and overbanks, a determination of a composite value for the cross section is necessary (Chow 959). The choice of a channel compositing method is very important in stream restoration design where large differences exist in bank and bed roughness. While the following example uses the Lotter method, other methods, such as the equal velocity method and the conveyance method, can also be used. Example problem: Composite Manning s n value Problem: Determine a composite n value for the cross section illustrated in figure 6 5 at the given depth of flow. Assume that this channel is experiencing a 6,094 cubic feet per second flow, with 5,770 cubic feet per second in the main channel and the remainder on the right overbank. The mean velocity in the main channel is. feet per second and on the overbank, 0.55 foot per second. The channel slope is , and a fairly regular profile of clay and silt is observed. The channel is relatively straight and free of vegetation up to a stage of 0 feet. Above that level, both banks are lined with snags, shrubs, and overhanging trees. The right overbank is heavily timbered with standing trees up to 6 inches in diameter with significant forest litter. In stream work, the convention is that the Figure 6 5 Cross-sectional dimensions While there has been considerable research on estimating roughness coefficients since 959, flood plain and channel n values are still challenging to determine. In practice, to a large extent the selection of Manning s n values remains judgement based. 5 ft 50 ft 40 ft ft 60 ft 80 ft 8 ft 6 (0 VI NEH, August 007)

19 left bank is on the left when looking downstream. See figures 6 6 and 6 7 where the photos are taken at a lower stage (Barnes 967). Solution: To determine the composite Manning s n value, the inchannel and overbank n values must first be determined. The solution will first estimate n values using reference materials, then this solution will compare this estimate with the value calculated from Manning s equation. Roughness estimates can be found in NEH 5, Hydraulics, supplement B by Cowan (956). Arcement and Schneider (990) extended this body of work. Both methods estimate a base n value for a straight, uniform, smooth channel in natural materials, then modifying values are added for channel irregularity, channel cross-sectional variation, obstructions, and vegetation. After these adjustments are totaled, an adjustment for meandering is also available. For the channel below 0 feet, the bed material is silty clay. Arcement and Schneider (990) show base n values for sand and gravel. For firm soil, their n value ranges from 0.05 to 0.0. Cowan (956) shows a base n of 0.00 for earth channels. Richardson, Simons, and Lagasse (00) shows 0.00 for alluvial silt and 0.05 for stiff clay. A reasonable assumption could be 0.04 for the channel below 0 feet of depth. For the remainder of the channel, above 0 feet of depth to top of bank at 0 feet, the effects of vegetation must be added in. The channel is then divided into three pieces: a lower channel, an upper channel, and a right overbank. Other breakdowns of this cross section are possible. For the lower channel a base n value of 0.04 is assumed. Referring to Cowan (956) in NEH 5, supplement B, a can be added for minor irregularity and a addition for a shifting cross section. This gives a total n value for the lower channel of Figure 6 6 Looking upstream from left bank Figure 6 7 Looking downstream on right overbank (0 VI NEH, August 007) 6

20 For the upper channel, the area above the lower 0 feet of flow depth and excluding the right overbank, the base n value is 0.04, a minor irregularity addition of 0.005, a addition for a shifting cross section, a minor obstruction addition of 0.00, and a medium vegetation addition of 0.00 can be selected. This gives a total n value for the upper channel of For the overbank, a base n (from the overbank soil) is needed. Based on site-specific observations, it was found that the soil is slightly more coarse than that of the main channel, n = Again from NEH 5, supplement B, Cowan (956) a minor irregularity addition of 0.005, a shifting cross section addition of 0.005, an appreciable obstruction addition of 0.00, and a high vegetation addition of 0.00 can be selected. This gives a total n value for the overbank of To obtain composite roughness, use the method of Chow (959), whereby a proportioning is done with wetted perimeter (P) and hydraulic radius (R): As follows: n = N PR 5 PN R nn 5 N P A R n x s part (ft) (ft ) (ft) Lower channel Upper channel Right overbank Total channel 59, Total x s 48,90.70 (eq. 6 ) n chan = ( )( ) ( ) = Discussion: The difference in Manning s n initially appears to be cause for concern. However, it does illustrate three important points. First, this process is subjective, and two equally capable practitioners may arrive at different results. Second, Manning s equation is for uniform flow. Differences in measured and calculated n values should be attributed to the uncertainty in choosing appropriate values to account for various factors associated with roughness. Manning s equation by itself can provide an estimate, but it cannot precisely determine roughness when the flow is not uniform. Third, an uncertainty analysis is recommended for hydraulic analysis. As documented in Barnes (967), the USGS backwater calculations determined the channel n value to be and the right overbank n value to be In contrast to this example, Barnes calculated roughness using energy slope, rather than water surface slope and also included expansion and contraction losses. Example problem: Manning s n value for a sand-bed channel Problem: Determine the n value for a wide, sand channel with the following cross section (fig. 6 8). Assume a discharge of 4,00 cubic feet per second, a thalweg depth of 5 feet, : side slopes and a fairly straight, regular reach. Assume a slope of 0.00 and a sandy bottom with a D 50 of 0. millimeter. Using equation 6 the composite roughness is: n = ( 94)( 6. 9) n = ( 48)(. 70) ( 65 )( 8. 8 ) ( 89 ) ( 4.) This value can be compared to a value calculated with Manning s equation as follows. 5 Figure 6 8 Sand channel cross section 5 ft n = 486. Q AR S ft 5 ft ft 6 4 (0 VI NEH, August 007)

21 Solution: Roughness in sand channels is highly dependent on the channel bedforms, and bedforms are a function of stream power and the sand gradation. Arcement and Schneider (990) show suggested n values for various D 50 values with the footnote that they apply only for upper regime flows where grain roughness is predominant. For a D 50 of 0. millimeter, this reference suggests a 0.07 n value. However, it is important to assess the regime of the flow. A figure from Simons and Richardson (966) (also in Richardson, Simons, and Lagasse 00 and Arcement and Schneider 990) is shown as figure 6 9. Given stream power and median fall diameter, the flow regime may be estimated, as well as the expected bedform and roughness range. Stream power may be calculated from where gamma is unit weight of water, Q is discharge, and S f is the energy slope. Assuming the energy slope is nearly the same as the bed slope, then: Ω = ( 6. 4 lb/ft )( 400 ft /s)( 0. 00) = lb/s (per ft of channel length) For figure 6 9, stream power per cross-sectional area is needed. The flow area for the given cross section is 554 ft, so the stream power is 0.60 pounds per second per square foot (per foot of channel length). Reading figure 6 9, with a D 50 of 0. millimeter, the flow is in the upper regime, but close to the transition. This would support an n value of 0.07, particularly if bedforms are present. Figure 6 9 Plot of flow regimes resulting from stream power vs. median fall diameter of sediment Stream power, 6 RS w V (ft-lb/s/ft ) Upper regime Lower regime Transition Upper flow regime Plane bed Antidunes Standing waves Breaking waves Chute and pools Lower flow regime Ripples Dunes (0.00 n 0.0) (0.00 n 0.0) (0.0 n 0.00) (0.08 n 0.05) (0.08 n 0.08) (0.00 n 0.040) Median grain size (mm) (0 VI NEH, August 007) 6 5

22 Figure 6 0 (Arcement and Schneider 990) indicates the general bedforms for increasing stream power. The anticipated bedform is a plane bed, and figure 6 9 suggests an n value between 0.00 and 0.0 for plane beds. The presence of breaking waves over antidunes would raise the roughness estimate to between 0.0 to 0.0. Finally, an estimate may be calculated with the Strickler formula (Chang 988; Chow 959) that relates n value to grain roughness. So, for a plane bed it should give a good estimate: 6 n = 0 089( 50 ) or. D with D 50 in feet (eq. 6 ) 6 n = ( 50 ). D with D 50 in meters (eq. 6 ) Since the D 50 is 0. millimeter, the calculated n value is 0.0, which agrees with figure 6 9 results for plane beds. Arcement and Schneider (990) show n = 0.0 for a D 50 of 0. millimeter, and this calculation is close to the transition range. Considering all of the above, information supports a roughness selection between 0.0 to If field observations support the plane bed assumption, a value from the low end of this range should be selected. If antidunes are present, a value from the high end of this range would be reasonable. Example problem: Manning s n value for a gravel-bed channel Problem: Determine the n value for a wide, gravel-bed channel with a D 50 of 0 millimeters. Assume a fairly straight, regular reach. Assume minimal vegetation and bedform influence. Solution: Since the grain roughness is predominant, the Strickler formula can be used. 6 n = ( 50 ). D for D 50 in meters This results in an estimated n value of 0.0. It should be noted that this estimate does not take into account many of the factors which influence roughness in natural channels. As a result, a estimate made with Strickler s equation is often only used as an initial, rough estimate or as a lower bound. Figure 6 0 General bedforms for increasing stream power Bedform Plane bed Water surface Ripples Dunes Transition Plane bed Standing waves and antidunes Bed Resistance to flow (Manning s roughness coefficient) Lower regime Transition Upper regime Stream power 6 6 (0 VI NEH, August 007)

23 (d) Friction factor As with Manning s n value and the Chézy C, the friction factor, f, is a roughness coefficient in a velocity equation, namely, the Darcy-Weisbach equation. Originally developed for pipe flow, the equation adapted for flow in open channels is: grs V = f with f being dimensionless. (eq. 6 4) grs Alternatively, f = 8 V (eq. 6 5) In 96, the ASCE Task Committee on Friction Factors in Open Channels recommended the preferential use of the Darcy-Weisbach friction factor over Manning s n (Simons and Sentürk 99). While Manning s equation remains the most used equation in practice, a comparison between the two is an illustrative exercise. The equation, applicable for steady uniform flow, is a balance of downstream gravitational force and upstream boundary resistance forces. The relationship between Manning s n and Chézy C is (Hey 979, English units): d C f = = ng g d = hydraulic depth (eq. 6 6) To apply the velocity equation, the friction factor must be determined. As has often been discussed by researchers (Raudkivi 990; Thorne, Hey, and Newson 00), the vertical velocity profile can often be assumed to be logarithmic with distance from the bed. For sand and gravel channels, where the relative roughness (flow depth/bed-material size) exceeds 0, this relationship holds. For use in gravel-bed streams, with width-to-depth ratios greater than about 5, Hey (979) derived the following (see also Thorne, Hey, and Newson 00): or f 0. 5 ar =. 0 log (SI units) (eq. 6 7). 5D 84 8 ar 5 75 f =. log (English units) (eq. 6 8). 5D 84 R = hydraulic radius D 84 = bed-material size for which 84 percent is smaller The dimensionless a is given by (Thorne, Hey, and Newson 00): 0. 4 R a =. dmax d max = maximum flow depth (eq. 6 9) The coefficient a varies from. to.46 and is a function of channel cross-sectional shape. For channels in which the width-to-depth ratio exceeds, the maximum flow depth is valid in the above equation. Otherwise, the value in the denominator should be the distance perpendicular from the bed surface to the point of maximum velocity. This formula for determining f may be used in gravel-bed riffle-pool streams in the riffle section, where flow is often assumed to be uniform. In general, the D 84 is calculated based on a sample taken at the riffle section. The Limerinos equation can also be used to determine the friction factor. ( ) R n = r (eq. 6 0) log D84 R = hydraulic radius, in ft D 84 = particle diameter, in ft, that equals or exceeds that of 84 percent of the particles This equation was developed from samples taken from large United States rivers with bed materials ranging from small gravel to medium size boulders. This equation has been shown to work well on sand-bed streams with plane beds. (e) Accounting for velocity distributions in water surface profiles Actual velocities in a cross section are distributed from highest, generally in the center at a depth that is some small proportion beneath the surface, to much (0 VI NEH, August 007) 6 7

24 lower values in overbanks and at flow boundaries (fig. 6 ). A velocity meter measures velocities related to the vertical flow area close to the instrument. This elementary phenomenon is responsible for the fact that an average cross-sectional velocity cannot provide a precise measure of the kinetic energy of the flow; the alpha and beta coefficients therefore are needed as modifiers. When the flow velocity in a cross section is not uniformly distributed, the kinetic energy of the flow, or velocity head, is generally greater than V /g, where V is the average velocity. The true velocity head may be approximated by multiplying the velocity head by alpha (α), the energy coefficient. Chow (959) stated that experiments generally place alpha between.0 and.6 for fairly straight prismatic channels. The nonuniformity of velocity distribution also influences momentum calculations (as momentum is a function of velocity). Beta (β) is the momentum coefficient that Chow indicates varies from.0 to. for fairly straight prismatic channels. Beta, also called the Boussinesq coefficient, is also described in Chow (959). Both coefficients may be calculated by dividing the flow area into subareas of generally uniform velocity distribution. v i A i α (eq. 6 ) V A total v i A β V A i total (eq. 6 ) However, for natural channels, the calculation is better made using conveyance. HEC RAS uses the following formulas: Figure 6 Flow velocities for a typical cross section K i A i α K A total total K i A i β K total A total (eq. 6 ) (eq. 6 4) Every cross section is only a two-dimensional slice of a three-dimensional reality. Cross sections change along the stream profile, inevitably setting up transverse velocity vectors, and the flow is induced into a roughly spiral motion. This flow behavior leads to point bars, pools and riffles, meandering patterns, and flood plains. Further information on the velocity and shear in the design of streambank protection in bends is given in NEH654.4, Stabilization Techniques. (f) Determining the water surface in curved channels Water surface profiles as computed by HEC RAS assume a level water surface in each cross section. This is not the case in a curved channel. However, the water surface calculated by HEC RAS is valid along the centerline of the flow. Generally, HEC RAS can account for the friction and eddy losses caused by a bend so that the water surface computed upstream would be correct. However, the super-elevated water surface in the bend itself must be calculated separately. The following formula is often used for estimating super-elevation in a water surface. Z = bv gr c (eq. 6 5) V = average channel velocity (ft/s) b = channel top width (ft) g = gravitational acceleration (. ft/s ) r c = radius of curvature of the channel (ft) Z = super-elevation in ft from bank to bank, so the amount added to or subtracted from the centerline elevation would be half that. A factor of safety of.5 is generally applied. 6 8 (0 VI NEH, August 007)

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